ComAlgebra

Commutative Algebra - MTH 423 (for UG + iPhD)

January - April 2017

Instructor : Dr. Anupam Singh

Schedule : Lecture : Tuesday at 9am

Lecture : Thursday at 11pm

Tutorial : Friday at 2pm

Evaluation : Test I - 20%

Mid-sem examination- 30%

Test II- 20%

End-sem examination- 30%

Prerequisite : motivation to learn Algebra

Goal of the course : This topic prepares necessary background for the subject of Algebraic Number Theory and Algebraic Geometry.

Proposed content : Rings, Modules, Localisation, Tensor product, Noetherian and Artinian rings, Integral dependence, Dedekind domains, Noether Normalisation, Hilbert’s Nullstellensatz, Affine algebra over a field.

References :

  1. Steps in Commutative Algebra, R. Y. Sharp

  2. Introduction to Commutative Algebra, Atiyah and Macdonald

Further Reading : One can take up the following subjects from here on: Algebraic Geometry, Algebraic Groups, Elliptic Curves (which is also applied in cryptography), Commutative algebra, Non-commutative associative algebras, Algebraic Number Theory, Computational commutative algebra and Algebraic Geometry, etc.

  • 5 January 2017 - Solving equations, Rings and Modules

  • 6 January 2017 - Examples of rings and modules Assignment I

  • 10 January 2017 - Ideals, prime ideals and maximal ideals

  • 12 January 2017 - (generalised) Euclidean algorithm for polynomials, Example of group ring

  • 13 January 2017 - Tutorials

  • 17 January 2017 - Affine space, Closed sets, Affine algebraic variety

  • 19 January 2017 - Zariski Topology

  • 20 January 2017 - Tutorials

  • 24 January 2017 - Irreducible varieties, Noetherian topological spaces and irreducible components

  • 26 January 2017 - Holiday

  • 27 January 2017 - Tutorials

  • 31 January 2017 - Noetherian Modules, Noetherian rings

  • 2 February 2017 - Hilbert Basis Theorem Assignment II

  • 3 February 2017 - Test I

  • 7 February 2017 - Field of fractions, Localisation of a ring

  • 9 February 2017 - Localisation of a module, examples

  • 10 February 2017 - Tutorial

  • 14 February 2017 - Local property

  • 16 February 2017 - Local Rings

  • 17 February 2017 - Tutorial

  • 23 February 2017 Mid Sem Exam

  • 28 February 2017 - Integral extension, examples

  • 2 March 2017 - Integral closure, integrally closed, examples

  • 3 March 2017 - Tutorial

  • 7 March 2017 - Algebraic independence

  • 9 March 2017 - Noether's Normalisation Lemma

  • 10 March 2017 - Tutorial Assignment III

  • 14 March 2017 - Hilbert's Nullstellensatz (version I)

  • 16 March 2017 - Hilbert's Nullstellensatz (version II), Co-ordinate ring, Affine k-algebra

  • 17 March 2017 - Tutorial Assignment IV

  • 21 March 2017 - variety, morphism of variety, morphism inducing algebra homomorphism at coordinate level

  • 23 March 2017 - Examples

  • 24 March 2017 - Tutorial

  • 28 March 2017 - Holiday

  • 30 March 2017 - Tensor Product of modules

  • 31 March 2017 - Test II

  • 4 April 2017 - Examples of tensor product Assignment V

  • 6 April 2017 - Base change

  • 7 April 2017 - Tutorial

  • 11 April 2017 - Story of Algebraic Groups

  • 13 April 2017 - Guest lecture by Vivek Mallick "Intersection"

  • 14 April 2017 - Holiday

  • 28 April 2017 - End Sem Exam